Formulations of the Elastodynamic Equations in Anisotropic and Multiphasic Porous Media from the Principle of Energy Conservation

TL;DR

This paper introduces a novel energy conservation-based methodology for formulating elastodynamic equations in anisotropic, multiphasic porous media, providing clearer physical insights and broader applicability than classical variational methods.

Contribution

The paper develops a new approach to derive elastodynamic equations directly from energy conservation, applicable to complex media, and revisits interface conditions with this framework.

Findings

Derived elastodynamic equations using energy conservation in anisotropic media

Validated formulations by comparing with classical principles

Extended interface conditions to explain reflection and refraction laws

Abstract

Elastodynamic equations have been formulated with either Newton's second law of motion, Lagrange's equation, or Hamilton's principle for over 150 years. In this work, contrary to classical continuum mechanics, a novel strategic methodology is proposed for formulating general mechanical equations using the principle of energy conservation. Firstly, based on Hamilton's principle, Hamilton's equations, Lagrange's equation, and the elastodynamic equation of motion are derived in arbitrarily anisotropic and multiphasic porous elastic media, for the first time. Secondly, these equations are all formulated using the principle of energy conservation for the related media. Both formulation results using the two kinds of principles are compared and validated by each other. The advantages of our methodology lie in that, the elastodynamic equation of motion, Lagrange's equation, and Hamilton's equations in continuum mechanics are directly formulated using a simple constraint of energy conservation without introducing variational concepts. It is easy to understand and has clear physical meanings. Our methodology unlocks the physics essences of Hamilton's principle in continuum mechanics, which is a consequence of the principle of energy conservation. Although the linear stress-strain constitutive relation is considered, our methodology can still be used in a nonlinear dynamical system. The methodology also paves an alternative way of treating other complex continuous dynamical systems in a broad sense. In addition, as an application, the continuity conditions at various medium interfaces are also revisited and extended using our proposed approach, which explains the law of reflections and refractions.